Question

"What is the amount of 10 equal annual deposits that can provide five annual withdrawals when a first withdrawal of $5,000 is made at the end of year 11 and subsequent withdrawals increase at the rate of 8% per year over the previous year's withdrawal, if the interest rate is: (a) 7% compounded annually? (b) 6% compounded annually?

In detail show and explain all steps

Answer 1

Answer (a):

Interest rate = 7% compounded annually

First let us calculate PV at the start of year 11 (at the end of year 10) of annual withdrawals growing at 8% for 5 years

PV = (P / (r - g)) * (1 - ((1+g) / (1+ r)) ⁿ) where P first withdrawal, r = rate of interest, g = Growth rate and n = number of years

PV =(5000 / (7% - 8%)) * (1 - ((1 + 8%) / (1 + 7%)) ⁵

= $23,805.3059

Now we need to get required annual equal installments for 10 years earning 7% rate compounded annually which results in future value = $23,805.3059

We will use excel function PMT to get the same:

PMT (rate, nper, pv, fv, type)

= PMT (7%, 10, 0, -23805.3059, 0)

=$1,722.97

Required annual deposit = $1,722.97

Answer (b):

Interest rate = 6% compounded annually

PV at start of year 11 =(5000 / (6% - 8%)) * (1 - ((1 + 8%) / (1 + 6%)) ⁵

= $24,491.8535

Now we need to get required annual equal installments for 10 years earning 6% rate compounded annually which results in future value = $24,491.8535

PMT (rate, nper, pv, fv, type)

= PMT (6%, 10, 0, -24491.8535, 0)

=$1,858.15

Required annual deposit = $1,858.15